I believe that I could try to compute the same integral with limits from $-\infty$ to $\infty$ using residue on a half circle and then let the radius tend off to infinity, and once I have that value I ...

$I = \int^{2\pi}_0 \dfrac{d\theta}{2 - \cos \theta}$
This is straight from a book I'm reading, which suggests to convert $\cos\theta$ into $0.5(z+1/z)$ and then solve the integral on the unit circle. ...

I am trying to find an integral:
$$\int_{-\infty}^{+\infty}\frac{e^{-\sqrt{(x^2 + 1)}}}{(x^2 + 1)^2}\,\mathrm dx$$
I went about applying contour integral over a semicircle with diameter along $ x = ...

Is it possible to calculate in closed-form the integral $\int_{-\infty}^{+\infty}\mathrm{sinc}(\sqrt{1+x^4})\,dx$ (sinc being the cardinal sine, $\sin(x)/x$)?
The function is everywhere defined (all ...

$$\int_{\gamma(0;2)}\frac{e^{i\pi z/2}}{z^2-1} \, dz$$
Using the residue calculus i got $$-2\pi$$But the answer is $$=i$$ I must be wrong at this, but shouldn't the answer have $\pi$ at least since ...

The problem is $\int_{0}^{\infty} \frac{\sqrt{x}}{x^2+2x+5}dx$
I replace x with z, and did some algebra, but the solution was rather nasty. it contains exponential and arctan such and such. However, ...

I need a little help with the following problem. I've tried many ways, but i didnt succeed. I think there needs to be a trick or something, some transformation. The task is to find the residue of the ...

Compute the residues at all the singularities of the function $tanh(z)$, and compute the integral
$\int_C tanh(z)$ where C is the circle of radius 12 centred at $z_0 = 0$.
attempt:singularities are ...

I need to find the residue of $e^{\frac 1{1-z}}$ using Laurent series. How would I manipulate the function to make it easier? I need to find all singularities and the corresponding residues. I believe ...

I came across a theorem used to calculate residues of rational functions that states that if f and g are analytic functions at $z_k$ and $g'(z_k)$ isn't 0, then the residue of $f(z)/g(z)$ at $z_k$ is ...

Respected All.
I was studying residue theory where I came accross the following problem
"If $f$ be analytic in the simply connected domain $D$ and $z_1, z_2$ are two distinct complex point lying in ...

Let $R(x)$ be rational function. It is any general method to calculate $\int_{0}^{\infty}R(x) \log(x)dx$ ? I can do it in special cases, but I am looking for a general method. What should be a minimal ...

I want to integrate $$\int_{0}^{\infty}\frac{1}{(1+x)^5}dx$$ by the method of residue, but I have done only problems of simple poles, but this is a problem of fifth order pole. So I am stuck in it. ...

Based on wiki, the residues of $\Gamma$ at non positive integers are given by:
$$\text{Res}\left ( \Gamma(z),z=-n \right )=\frac{(-1)^{n}}{n!}.$$
I have been trying to find residue for $\Gamma^{2}$ ...

Let $f$ be an entire function on the complex plane, with Taylor's expansion around zero as $f(z) = \sum_{k=0}^{\infty}c_{k}z^{k}$. Let $N(r)$ be the number of zeroes of $f$ in $D(0, r)$. Show that for ...